Assignment (MComm)--University of Stellenbosch, 2005. / Bibliography. Also available via the Internet.

Thesis (Ph. D.)--University of Texas at Dallas, 2006. / Includes vita. Includes bibliographical references (leaves 92-93).

In this thesis, we review the basic notions related to univariate regular variation and study some fundamental properties of regularly varying random variables. We then consider the notion of regular variation in the multivariate case. After collecting some results from multivariate regular variation for random vectors with values in $\mathbb{R}_{+}^{d}$, we discuss its properties and examine several examples of multivariate regularly varying random vectors such as independent and identically distributed random vectors, fully dependent random vectors and other models. We also present the elements of univariate and multivariate extreme value theory and emphasize the connection with multivariate regular variation. Some measures of extreme dependence such as the stable tail dependence function and the Pickands dependence function are presented. We end the study by conducting a data analysis using financial data. In the univariate case, graphical tools such as quantile-quantile plots, mean excess plots and Hill plots are used in order to determine the underlying distribution of the univariate data. In the multivariate case, non-parametric estimators of the stable tail dependence function and the Pickands dependence function are used to describe the dependence structure of the multivariate data.

The Frechet distribution was applied to a set of earthquake data in order to test its validity as a practical alternative distribution for extreme value data. It was concluded that the Frechet distribution was the best model representing that data set. Also, a Poisson model of occurrence could not be rejected for that data set. The combination of these two models resulted in a closed form unconditional extreme value distribution which was developed analytically. The appropriate statistical tests and sensitivity analyses were performed on the obtained model. / Applied Science, Faculty of / Mechanical Engineering, Department of / Graduate

Modern portfolio theory (MPT) is an investment theory which was introduced by Harry Markowitz in 1952 and describes how risk averse investors can optimize their portfolios. The objective of MPT is to assemble a portfolio by maximizing the expected return given a level of market risk or minimizing the market risk given an expected return. Although MPT has gained popularity over the years it has also been criticized for several theoretical and empirical shortcomings such as using variance as a measure of risk, measuring the dependence with linear correlation and assuming that returns are normally distributed when in fact empirical data suggests otherwise. When moving away from the assumption that returns are elliptical distributed, for example normally distributed, we can not use linear correlation as a measure of dependence in an accurate way. Copulas are a flexible tool for modeling dependence of random variables and enable us to separate the marginals from any joint distribution in order to extract the dependence structure. The objective of this paper was to examine the applicability of a copula-CVaR framework in portfolio optimization compared to the traditional MPT. Further, we studied how the presence of memory, when calibrating the copulas, affects portfolio optimization. The marginals for the copula based portfolios were constructed using Extreme Value Theory and the market risk was measured by Conditional Value at Risk. We implemented a dynamic investing strategy where the portfolios were optimized on a monthly basis with two different length of rolling calibration windows. The portfolios were backtested during a sample period from 2000-2016 and compared against two benchmarks; Markowitz portfolio based on normally distributed returns and an equally weighted, non optimized portfolio. The results demonstrated that portfolio optimization is often preferred compared to choosing an equally weighted portfolio. However, the results also indicated that the copula based portfolios do not always beat the traditional Markowitz portfolio. Furthermore, the results indicated that the choice of length of calibration window affects the selected portfolios and consequently also the performance. This result was supported both by the performance metrics and the stability of the estimated copula parameters.

by Leung Tsun Ip. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2006. / Includes bibliographical references (leaves 72-78). / Abstracts in English and Chinese. / Chapter 1. --- Introduction --- p.1 / Chapter 2. --- Literature Review --- p.6 / Chapter 2.1 --- Development of estimation of Value-at-Risk (VaR) --- p.6 / Chapter 2.2 --- Methods to evaluate VaR --- p.9 / Chapter 2.2.1 --- Non-paremetric Method --- p.9 / Chapter 2.2.2 --- Semi-parametric Method --- p.11 / Chapter 2.2.3 --- Parametric Method --- p.12 / Chapter 3. --- Extreme Value Theory (EVT) --- p.16 / Chapter 3.1 --- Introduction of Extreme Value Theory (EVT) --- p.16 / Chapter 3.1.1 --- Block Maxima Approach --- p.18 / Chapter 3.1.2 --- Peaks over Threshold (POT) Approach --- p.21 / Chapter 3.1.3 --- Comparison between Block Maxima and POT Approach --- p.22 / Chapter 3.2 --- Numerical Illustration --- p.23 / Chapter 3.2.1 --- Data --- p.23 / Chapter 3.2.2 --- Diagnosis --- p.24 / Chapter 4. --- Dynamic Extreme Value Theory (DEVT) --- p.29 / Chapter 4.1 --- Theoretical Framework of DEVT --- p.29 / Chapter 4.2 --- Estimation of Parameters --- p.32 / Chapter 4.3 --- Determination of Threshold Level --- p.37 / Chapter 4.4 --- Estimation of zq --- p.44 / Chapter 5. --- Backtesting and Time Aggregation --- p.49 / Chapter 5.1 --- Backtesting DEVT --- p.49 / Chapter 5.2 --- Time Aggregation --- p.55 / Chapter 6. --- Case Study: China Aviation Oil Singapore (CAO) Incident --- p.61 / Chapter 6.1 --- Background Information --- p.61 / Chapter 6.2 --- Data Analysis --- p.63 / Chapter 6.3 --- Suggestion --- p.68 / Chapter 7. --- Discussion --- p.71 / References --- p.72 / Chapter A. --- Appendix --- p.79

This study attempts to quantify the shocks to a banking network and analyze the transfer of shocks through the network. We consider two sources of shocks: external shocks due to market and macroeconomic factors which impact the entire banking system, and idiosyncratic shocks due to failure of a single bank. The external shocks will be estimated by using two methods: (i) non-parametric simulation of the time series of shocks that occurred to the banking system in the past, and (ii) using the extreme value theory (EVT) to model the tail part of the shocks. The external shocks we considered in this study are due to exchange rate and treasury bill rate volatility. Also, an ARMA/GARCH model is used to extract iid residuals for this purpose. In the next step, the probability of the failure of banks in the system is studied by using Monte Carlo simulation. We calibrate the model such that the network resembles the Canadian banking system.

This study attempts to quantify the shocks to a banking network and analyze the transfer of shocks through the network. We consider two sources of shocks: external shocks due to market and macroeconomic factors which impact the entire banking system, and idiosyncratic shocks due to failure of a single bank. The external shocks will be estimated by using two methods: (i) non-parametric simulation of the time series of shocks that occurred to the banking system in the past, and (ii) using the extreme value theory (EVT) to model the tail part of the shocks. The external shocks we considered in this study are due to exchange rate and treasury bill rate volatility. Also, an ARMA/GARCH model is used to extract iid residuals for this purpose. In the next step, the probability of the failure of banks in the system is studied by using Monte Carlo simulation. We calibrate the model such that the network resembles the Canadian banking system.

MV is the traditional method to treat international portfolio selection problems, which bases its theory on the assumption of Normal Distribution. However, during economy recession the portfolio return turns out to be a fat tail distribution. Therefore, in this sense, we explore Roy’s SF criterion and apply the extreme theory to the historical data. We demonstrate how such portfolios would perform during the Asian Crisis, IT Bubble Bust and the Financial Crisis separately. We also compare the SF portfolio’s performance to the MV portfolio’s performance, therefore to check, SF and MV portfolio, which will outperform during bust and boom of the economy. The Asian Crisis was marked with great currency devaluation and lower currency return on equity. The Dot.Com Bubble Busts was known for its sharp plummet in the stock market, while, the Financial Crisis was known as the large falls in the US stock market and elsewhere. They are the extreme events of the world capital markets, which in some way contribute to the non-normal distribution. Simulated results over the 1997-2010 period which include six busts and booms: the Asian Crisis, period after Asian Crisis, IT Bubble Bust, period after IT Bubble Bust, The Financial Crisis and period after The Financial Crisis, indicate that SF portfolio outperforms MV portfolio during most of the times, this result is especially obvious for Indonesian and Thailand.

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